Factor the following expression: $9$ $x^2+$ $10$ $x+$ $1$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(1)} &=& 9 \\ {a} + {b} &=& & & {10} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $9$ and add them together. The factors that add up to ${10}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${9}$ $ \begin{eqnarray} {ab} &=& ({1})({9}) &=& 9 \\ {a} + {b} &=& {1} + {9} &=& 10 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 +{1}x +{9}x +{1} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 +{1}x) + ({9}x +{1}) $ Factor out the common factors: $ x(9x + 1) + 1(9x + 1) $ Notice how $(9x + 1)$ has become a common factor. Factor this out to find the answer. $(9x + 1)(x + 1)$